Fungrim entry: 503d4d

\operatorname{atan}\!\left(x\right) - \operatorname{atan}\!\left(y\right) = \operatorname{atan}\!\left(\frac{x - y}{1 + x y}\right)

Assumptions:

x \in \mathbb{C} \,\mathbin{\operatorname{and}}\, y \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|x\right| \lt 1 \,\mathbin{\operatorname{and}}\, \left|y\right| \lt 1

Alternative assumptions:

x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R} \,\mathbin{\operatorname{and}}\, x y \gt -1

TeX:

\operatorname{atan}\!\left(x\right) - \operatorname{atan}\!\left(y\right) = \operatorname{atan}\!\left(\frac{x - y}{1 + x y}\right)

x \in \mathbb{C} \,\mathbin{\operatorname{and}}\, y \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|x\right| \lt 1 \,\mathbin{\operatorname{and}}\, \left|y\right| \lt 1

x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R} \,\mathbin{\operatorname{and}}\, x y \gt -1

Definitions:

Fungrim symbol	Notation	Short description
`Atan`	$\operatorname{atan}\!\left(z\right)$	Inverse tangent
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("503d4d"),
    Formula(Equal(Sub(Atan(x), Atan(y)), Atan(Div(Sub(x, y), Add(1, Mul(x, y)))))),
    Variables(x, y),
    Assumptions(And(Element(x, CC), Element(y, CC), Less(Abs(x), 1), Less(Abs(y), 1)), And(Element(x, RR), Element(y, RR), Greater(Mul(x, y), -1))))

Topics using this entry

Inverse tangent